Damage constitutive model and mechanical properties of a concrete limestone composite after acid corrosion

Damage mechanics

Figure 1 delineates the intrinsic structural configuration of the CLC entity, encompassing natural microfractures, interstitial voids, the propagation of fractures induced by acidic corrosion, and the progressive degradation of microelements in response to escalating axial stress.

This manuscript categorizes three distinct damage variables. The initial type is the natural damage variable, represented by D_n, which is attributed to the preexisting microfractures intrinsic to the limestone constituents. The subsequent type is the damage variable associated with acidic corrosion, symbolized by D_c, stemming from the propagation of microfractures within the composite because of exposure to acidic corrosion. The final category is the stress-induced damage variable, denoted by D₂, which is generated by the application of triaxial compressive stress. Given the presence of natural fissures within the CLC, quantifying the initial damage presents a challenge, and the response under an acidic corrosion environment is further complicated. Consequently, this study quantifies the initial damage of the CLC material and the subsequent compound damage due to acidic corrosion by examining alterations in its macroscopic elastic modulus. When D₁ = D_c+D_n is read, the following equation is used:

where D₁ represents the composite damage variable, which encompasses the initial damage of the composite material as well as damage incurred due to acidic corrosion. E₀ denotes the elastic modulus of the limestone–concrete composite in its pristine state, which is devoid of natural fissures prior to any acid-induced damage. E_c signifies the elastic modulus of the limestone‒concrete composite subsequent to acid-induced damage, reflecting the degradation in mechanical properties.

According to Lemaitre’s strain equivalence hypothesis³², it can be assumed that the effective stress before acid corrosion is \(\:{\sigma}_{i}^{**}\) and that the effective stress after acid corrosion is \(\:{\sigma}_{i}^{*}\). Then, we have:

On the basis of Lemaitre’s strain equivalence theory and the generalized Hooke’s law, the effective axial stress of a CLC sample without natural fissures before acidic corrosion is defined as \(\:{\sigma}_{1}^{**}\). After acidic corrosion, the effective axial stress \(\:{\sigma}_{1}^{*}\) is then given by the following equation:

The internal damage variable D₂ of the limestone–concrete composite under loading is posited as the ratio of the destroyed infinitesimal elements, denoted by , to the total count of infinitesimal elements, denoted by . Therefore, D₂ is mathematically expressed as the quotient:

According to Lemaitre’s strain equivalence hypothesis³², the constitutive relationship between the macroscopic nominal principal stress \(\:{\sigma}_{i}\) and the effective principal stress \(\:{\sigma}_{i}^{*}\) of CLC after acidic corrosion under loading can be expressed as:

By substituting Eq. (5) into Eq. (2), the following expression is obtained:

Damage evolution equation

Determination of the element strength F

Upon reaching a stress level F that equals the strength S of a microelement, that microelement fails. Given that the strength S of the element adheres to a probabilistic distribution, the quantity of elements sustaining damage within any infinitesimal stress interval from F to F + dF is delineated as follows:

where p(F) is the probability density function of the element strength S:

where P(F) is the probability distribution function of the element strength S:

The classical Mohr‒Coulomb criterion encapsulates the essence of shear failure in materials, yet it does not account for the influence of the intermediate principal stress \(\:{\sigma}_{2}\)³³. Considering the impact of the intermediate principal stress \(\:{\sigma}_{2}\) in triaxial compression tests, this study employs the Drucker‒Prager criterion, which incorporates the intermediate principal stress, to formulate the microelement strength of the composite body. According to the Drucker‒Prager criterion under triaxial compression, the criterion can be mathematically expressed as follows:

where \(\:\phi\:\) refers to the internal angle of friction of the composite body, while \(\:\alpha\:\), f and\(\:{\rm\:K}\)denote the intrinsic material properties.

The relationships between three different physical quantities related to the stress state are combined:

Upon the synthesis of Eqs. (10)-(13), the resultant formulation is derived:

According to Hooke’s law:

Given that the circumferential stresses are identical, \(\:{\sigma}_{2}^{*}={\sigma}_{3}^{*}\), the equation changes:

By substituting Eqs. (5) and (16) into Eq. (14), the following equation can be obtained:

Micro-elements strength distribution F

Considering the composite subjected to acid corrosion as a continuous damage process under triaxial loading and assuming that the scale of the defects is sufficiently small, the following hypotheses are proposed: (i) Apart from the interface, the composite is isotropic at both the macroscopic and microscopic scales; (ii) Prior to the failure of the microelements, it obeys Hooke’s law, indicating that the microelements exhibit linear elastic characteristics. Considering that the strength of the microelement F follows a two-parameter Weibull probability distribution, the probability density function is given by:

where \(\:p\left(F\right)\) is the strength of the microelement, \(\:{F}_{0}\) is the scale parameter, and \(\:m\) is the shape parameter of the Weibull distribution.

By substituting Eq. (19) into Eq. (5), the following result can be obtained:

Determination of model parameters

Let the peak stress and peak strain be denoted by \(\:{\sigma}_{p}\) and \(\:{\upvarepsilon\:}_{p}\), respectively, and let \(\:{F}_{p}\) be the corresponding distribution variable. The relationships among these variables can be established as follows:

To derive the derivative of Eq. 20:

At the peak points of stress and strain, the following relationships between the parameters are established:

Substitute Eq. 24 into Eq. 23:

By simultaneously solving Eqs. 20–25, we can derive the following set of relationships:

Residual strength correction

When the deformation and failure of composite materials are analysed via the Weibull distribution function, the damage variable approaches 1 as the microelement strength tends toward infinity, leading to a situation where the stress decreases to zero as the strain approaches infinity. However, rock-like materials often exhibit a certain degree of residual strength after failure^34,35. For example, at a confining pressure of 0 MPa, the composite has a relatively low residual strength, which significantly increases at confining pressures of 5 and 10 MPa. Therefore, using the Weibull distribution function alone to establish the damage constitutive model for the composite may not adequately reflect the characteristics of residual strength. Consequently, a damage variable correction factor, denoted as δ (0 < δ ≤ 1), is proposed to correct the statistical damage constitutive model of the limestone–concrete composite. When δ = 1, the influence of residual strength is not considered; however, in this case, the damage variable correction factor δ lacks a clear physical interpretation. Thus, Eq. (20) can be rewritten as:

Formula (23) can be rewritten as:

Similarly, at the peak points of stress and strain, the following relationships between the parameters are established:

Functions 28, 29 and 30 can be used to obtain function parameters that take residual strength correction into account simultaneously:

Statistical damage constitutive model

By combining Eqs. 3, 6, and 20, the statistical damage constitutive model can be derived as follows:

By combining Eqs. 3, 6, and 28, the equation can be rewritten as follows: